In the quantum theory, states are naturally defined on general Cauchy hypersurfaces [1]}, [2]}, [3]}. The asymptotic symmetries are generated by operators that act on the physical Hilbert space and form a representation of the asymptotic symmetry algebra, up to possible central terms when these are algebraically permitted. One direct access to the quantum theory is based on the Hamiltonian formalism, which closely parallels the quantum structure. In the standard description, the classical state of the system is completely specified (including radiation if any) by the values of the dynamical variables on Cauchy hypersufaces, which asymptote spacelike infinity. A satisfactory formulation needs a specification of the fall-off of the phase space variables at spatial infinity, which should be such that the action and the variational principle are well-defined. The connection between symmetries and Hamiltonian generators is then given by standard theorems of classical mechanics. One finds in particular that the symmetries have a symplectic action and are captured by the moment map (possibly defined on the centrally extended algebra when central charges occur). This close parallel with the quantum formulation is one of reasons that make the Hamiltonian formalism instructive.
